Resultant Wave Equation

Q.

Two harmonic waves are represented in SI units by, $y_1(x,t)=0.2sin(x-3.0t)$ and $y_2(x,t)=0.2sin(x-3.0t+\varphi )$. What is the expression for the sum $y=y_1+y_{2}for\varphi =\frac{\pi }{2}$.

1. y = 8 sin (x - 3.0t)

2. y = 0.28 sin (x - 3.0t+$\frac{\pi }{4}$)

3. y = 0.2 sin (x - 3.0t+$\frac{\pi }{4}$)

4. y = 0.28 cos (x - 3.0t+$\frac{\pi }{4}$)

Q. Two travelling sinusoidal waves are described by the wave functions $y_1=10\sin [\pi (8x-1400t)]$
and $y_2=10\sin [\pi (8x-1400t-0.5)]$ where $x$, $y_1$, and $y_2$ are in metres and $t$ is in seconds. Which of the following option(s) is/are correct ?

1. Amplitude of the resultant wave is $10\sqrt{2}m$
2. Frequency of the resultant wave is $700$ $Hz$
3. Time period of the resultant wave is $0.1$ $s$
4. None of these

Q. Sources separated by $20\text{m}$ vibrate according to the equation $y_1=0.06\sin \pi t$ and $y_2=0.02\sin \pi t.$ They send out waves along a rod with speed $3\text{m/s}$. What is the equation of motion of a particle $12\text{m}$ from the first source and $8\text{m}$ from the second? Given $y_1,y_2$ are in metres.

1. $0.0173\sin \pi t-0.05\cos \pi t$
2. $0.0173\sin \pi t-0.017\cos \pi t$
3. $0.05\sin \pi t-0.0173\cos \pi t$
4. $0.05\sin \pi t-0.5\cos \pi t$